### Nuprl Lemma : assert-init-seg-nat-seq2

`∀f,g:finite-nat-seq().  (↑init-seg-nat-seq(f;g) `⇐⇒` ((fst(f)) ≤ (fst(g))) ∧ ((snd(f)) = (snd(g)) ∈ (ℕfst(f) ⟶ ℕ)))`

Proof

Definitions occuring in Statement :  init-seg-nat-seq: `init-seg-nat-seq(f;g)` finite-nat-seq: `finite-nat-seq()` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` pi1: `fst(t)` pi2: `snd(t)` le: `A ≤ B` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` init-seg-nat-seq: `init-seg-nat-seq(f;g)` finite-nat-seq: `finite-nat-seq()` pi1: `fst(t)` pi2: `snd(t)` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` prop: `ℙ` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` iff: `P `⇐⇒` Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` le: `A ≤ B` less_than': `less_than'(a;b)` not: `¬A` rev_implies: `P `` Q`
Lemmas referenced :  ble_wf bool_wf eqtt_to_assert eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot finite-nat-seq_wf assert-ble int_seg_wf nat_wf subtype_rel_dep_function int_seg_subtype false_wf subtype_rel_self le_wf assert-equal-upto-finite-nat-seq assert_wf equal-upto-finite-nat-seq_wf iff_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation sqequalHypSubstitution productElimination thin rename sqequalRule cut introduction extract_by_obid isectElimination setElimination hypothesisEquality hypothesis unionElimination equalityElimination equalityTransitivity equalitySymmetry independent_isectElimination dependent_pairFormation promote_hyp dependent_functionElimination instantiate cumulativity independent_functionElimination because_Cache voidElimination independent_pairFormation functionEquality natural_numberEquality functionExtensionality applyEquality lambdaEquality productEquality addLevel impliesFunctionality

Latex:
\mforall{}f,g:finite-nat-seq().    (\muparrow{}init-seg-nat-seq(f;g)  \mLeftarrow{}{}\mRightarrow{}  ((fst(f))  \mleq{}  (fst(g)))  \mwedge{}  ((snd(f))  =  (snd(g))))

Date html generated: 2017_04_20-AM-07_29_44
Last ObjectModification: 2017_02_27-PM-06_00_33

Theory : continuity

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